Basically what happens is that the body needs some time to start producing more red blood cells. If you stay at the higher level your body can't cope with the acute lack of oxygen and as a result you run the risk of altitude sickness, pulmonary edema, cerebral edema and death*. Altitude sickness can occur at heights above about 2500 m, so you don't even need to be all that high before the symptoms show.
The easiest way to prevent altitude sickness is to return to a lower level, preferably below the 2500 m level, but a level to which you are already acclimatized may suffice. Doing so gives the body some time to start making more red blood cells in response to the lower level of oxygen experienced at the high altitude. When you return to the higher altitude, your body has now adapted and you can function better. This process is often repeated several times, increasing the time spent at the higher altitude with each visit, so that the body can fully adapt.
Acclimatization processes are not a guarantee that you won't still get altitude sickness at the higher level, they just make it less likely. As you know already, there is a level above which people can not adapt, no matter how many times they go up and down, this is known as the death zone.
* what? you don't know who death is?